An object with a mass of #4 kg# is acted on by two forces. The first is #F_1= <4 N , 8 N># and the second is #F_2 = < 1 N, 7 N>#. What is the object's rate and direction of acceleration?

#vecF=vecF_1+vecF_2#

#vecF_1=〈4,8〉#

#vecF_2=〈1,7〉#

#vecF=〈4,8〉+〈1,7〉=〈5,15〉#

The magnitude of #vecF# is #∥vecF∥=∥〈5,15〉∥#

#=sqrt(5^2+15^2)=sqrt(25+225)=sqrt250=15.8N#

The rate of acceleration is #=F/m=15.8/4=3.95ms^(-2)#

The direction is #=arctan(15/5)=arctan3=71.6#º

The net force acting on the object is the vector sum of \( F_1 \) and \( F_2 \): \[ F_{\text{net}} = F_1 + F_2 = \left< 4 \, \text{N} + 1 \, \text{N}, \, 8 \, \text{N} + 7 \, \text{N} \right> = \left< 5 \, \text{N}, \, 15 \, \text{N} \right> \] To find the acceleration, we use Newton's second law, which states that \( F_{\text{net}} = m \cdot a \), where \( m \) is the mass of the object and \( a \) is its acceleration. Thus, the acceleration can be calculated as: \[ a = \frac{F_{\text{net}}}{m} = \left< \frac{5 \, \text{N}}{4 \, \text{kg}}, \, \frac{15 \, \text{N}}{4 \, \text{kg}} \right> = \left< 1.25 \, \text{m/s}^2, \, 3.75 \, \text{m/s}^2 \right> \] Therefore, the object's rate of acceleration is \( 1.25 \, \text{m/s}^2 \) in the \( x \)-direction and \( 3.75 \, \text{m/s}^2 \) in the \( y \)-direction.